As per the first component \(\tau _t\) , should one decide to do away with it (equivalent to take \(\tau _{t} \equiv 1 \; \forall t\) ), the outcome would be the base \(\mathsf {MEM}\) , as suggested by [1]}, with expectation targeting \(\mu \) ; to model \(\tau _t\) , one can resort to some smooth function of time [2]}, or, even, a Markov Switching approach as in [3]} based on the concept of average level of volatility within each regime.
In what follows, also in view of the multivariate specification below, we adopt a non parametric specification derived from [4]} in the context of the ACD model, where the only requirement is that \(\tau _{t}\)  is a sufficiently smooth function of time: we call it semi–nonparametric-\(\mathsf {MEM}\)  or \(\mathsf {SpMEM}\) .
For practical purposes, at the estimation stage, it takes the form:
\(\widehat{\tau }_{t} =\frac{\displaystyle \sum _{s = 1}^{T} x_{t}^{(\tau )} K \left( \frac{z_{t} - z_{s}}{h} \right) }{\displaystyle \sum _{s = 1}^{T} K \left( \frac{z_{t} - z_{s}}{h} \right)} \\\) 
